I've seen a lot of changes about how a half ton truck now can do what a half ton truck 20 years ago couldn't do. But to me the big thing still is, sure you 'might' make it without any problems, especially if you have no hills or heavy duty grades to climb or descend.
But when you get that mass rolling can a half ton truck safely STOP it!
I'd say start smaller and move up when you can get the 3/4 ton truck. Why risk everybody? It won't be fun if anything should happen.

I do live in New Jersey, which is relatively flat. I could see taking it to Western PA or down to DELMARVA for short weekends, but I most certainly won't be crossing the Rockies anytime soon. Worst case would be towing to Fl once per year and in immediate area on weekends during Spring, Summer, Fall

Also, with a TT that long be sure you have a QUALITY sway prevention set up, not a friction bar type. You'll thank me later.

I think you are going to be very close, or over your GVWR with a TT that size long before you hit your GCVWR.

For comparison, I have a 2015 ram 1500 with 1490 GVWR and 10,500lb tow capacity (which means nothing because the tongue will overload the truck before we get there).

I tow a Coachmen Freedom Express 292BHDS with a dry sticker weight of 6104, and a GVWR of 7740.

Inside/on the truck
People: 410
Dog: 35
Blue ox: 96

Total payload used before camper hitched: 541

1490-541 = 949 payload available before hitching the TT up.

At 12% desired tongue weight, max trailer weight under current setup could be as much as 7908 before overloading the truck. This still assumes yeti and food/clothes are inside the camper near axles. however the TT should never weigh more than 7740, so i would overload the TT before overloading the truck WHEN LOADED PROPERLY.

So I can get 667 pounds more trailer within specs if we travel with bikes, cooler, clothes and food inside the camper near or behind the axles and not all on the the tongue.

However, remember that things you put inside the TT result in 12% of their weight on the tongue. Things put in the truck result in 100% of their weight on the truck, so loading stuff into the TT is needed to be sure I don't overload the truck.

You can't put the kids and wife in the TT, so you'll have possibly more weight in the truck, reducing your available payload by more.

You'll have to do the math on what you put in the truck before you start attaching the TT, then see if you have enough left to accommodate 12% of your wet TT weight.

That was my point in the original post though - doesn't weight distribution remove some of that weight from the tongue and push it back onto the trailer axle and some forward onto the truck's front axle thus freeing up some of the payload capacity for people and cargo? The numbers I saw were 25% / 50% / 25%.

That's the only way I can see the 9,200 lbs. working. I can't imagine that Chevrolet would build a truck and tout its towing capacity at 9,200 lbs and 1. not be able to do it at all and 2. not be able to do it safely. They are opening themselves up to a huge lawsuit if someone were to load their truck up with a 9,200 lbs trailer and wreck it. My guess is the 9,200 lbs would be on the light side of the capabilities so that there's a built in margin of error. Especially, in today's litigious society.

I know I've seen a ton of posts about this, but none that addressed my specific question.

We are a family of 5 (Ages: 14, 10, 10). We are looking at a Surveyor 32BHDS that comes in dry at 6,866 lbs and has a GVWR of 9,534 lbs. The max tow capacity of the truck per the manual is 9,200 lbs. and requires weight-distribution hitch for any weights above 7,500 lbs. The hitch weight of the travel trailer is 734 lbs.

I do not intend to camp anywhere without hookups. Water weighs 8.34 lbs for gallon so that means 717 lbs. of that gross weight will never be realized, leaving my max trailer weight at ~8,800 lbs.

Truck Payload = 1,746 (according to door sticker)

Truck GVWR = 7,200 lbs.

GAWR Front = 3,950 lbs.

GAWR Rear = 3,950 lbs.

Max Hitch = 1,200 lbs with weight distributing hitch

My question is why doesn't any of the analysis that I see done online not reference the impacts the weight distributing hitch will have on payload, hitch weight, or trailer weight? Why doesn't anyone take this into account when evaluating the towing capacity of their vehicle. My owner's manual clearly states that weight distributing hitches are required for anything over 7,500 lbs.

My understanding is that a weight distributing hitch will take 25% off the hitch and put it back on the trailer, take 25% off the hitch and put it on the front axle of the truck, and leave 50% on the hitch directly. With that being said (towing dry), if my hitch weight of the truck is 734 lbs, doesn't that mean that 183.5 lbs will be added onto the weight of the trailer to bring it to 7,049 lbs, another 183.5 lbs will be added to the weight of the truck's front axle, and the remaining 367 lbs will be on the hitch?

Consumer Beware! The owners manual is for several models. Unless you ordered the truck with the max tow package, you could be way short on towing capacity. The size of the cab(2 door, quad, crew), bed, engine, transmission, axle ratios and trim packages. Example, 2 1500's same cabs, beds, engine and transmission but different axle ratios 3:10 vs 3.73. The 3.73 has a tow package. One would tow 5,000lbs the other 9,000lbs. Just because a pickup truck has a hitch, that doesn't mean it has a tow package. Being the vehicle is a lease, the owner is responsible for damages to the vehicle when you turn it in. The dealership can run your VIN number and print out the list of options on that vehicle and the available towing capacity.

That was my point in the original post though - doesn't weight distribution remove some of that weight from the tongue and push it back onto the trailer axle and some forward onto the truck's front axle thus freeing up some of the payload capacity for people and cargo? The numbers I saw were 25% / 50% / 25%.

That's the only way I can see the 9,200 lbs. working. I can't imagine that Chevrolet would build a truck and tout its towing capacity at 9,200 lbs and 1. not be able to do it at all and 2. not be able to do it safely. They are opening themselves up to a huge lawsuit if someone were to load their truck up with a 9,200 lbs trailer and wreck it. My guess is the 9,200 lbs would be on the light side of the capabilities so that there's a built in margin of error. Especially, in today's litigious society.

Make it really simple on yourself, go to a cat scale and do a proper weight of the truck and trailer
Lots of links on here for the steps to do that.
You will be very surprised to get the actual numbers in front of you